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KVPY Sample Paper KVPY Stream-SX Model Paper-14

  • question_answer
    Let the population of rabbits surviving at a time \[t\] be governed by the differential equation \[\frac{dp(t)}{dt}=\frac{1}{2}P(t)-200.\]If \[p(0)=100,\] then \[p(t)\]equals:

    A) \[400-{{300}^{\text{et/2}}}\]

    B) \[300-{{200}^{\text{et/2}}}\]

    C) \[600-{{500}^{\text{et/2}}}\]   

    D) \[400-{{300}^{e\,-\,\text{t/2}\,}}\]

    Correct Answer: A

    Solution :

    \[\frac{dp}{dt}=\frac{P-400}{2}\]
    \[\frac{dP}{P-400}=\frac{1}{2}dt\]
    \[In\left| P-400 \right|=\frac{1}{2}t+c\]
    at \[t=0,P=100\]
    In \[300=c\]
    \[In\left| \frac{P-400}{300}=\frac{t}{2} \right|\]
    \[\Rightarrow \]   \[\left| P-400 \right|=300\,{{e}^{\text{t/2}}}\]
    \[\Rightarrow \]   \[400-P=300\,{{e}^{\text{t/2}}}\,(as\,P\,<400)\]
    \[\Rightarrow \]   \[P=400-300\,{{e}^{\text{t/2}}}.\]


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